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It means that there is a point <math>a</math> in <math>[0,1]</math> such that <math>V</math> jumps from <math>0</math> to <math>1</math> right after the point. (It has to occur like that in order to fulfill the identity.) | It means that there is a point <math>a</math> in <math>[0,1]</math> such that <math>V</math> jumps from <math>0</math> to <math>1</math> right after the point. (It has to occur like that in order to fulfill the identity.) | ||
| − | So <math>f(x)= f(0) \ \forall x\in[0,a]</math> and <math>f(x)= f(0)-1 \ \forall x\in(a,1]</math> | + | So <math>f(x)= f(0) \ \forall x\in[0,a]</math> and <math>f(x)= f(0)-1 \ \forall x\in(a,1]</math>. Thus |
| − | <math>\int_0^1f(x)dx=af(0) + (1-a)(f(0)-1)</math> | + | <math>\frac{1}{3}=\int_0^1f(x)dx=af(0) + (1-a)(f(0)-1)</math> |
Revision as of 09:36, 22 July 2008
From the identity $ f(0)-(V_{0}^{x})^{1/2} = f(x) $ $ \forall x\in[0,1] $ we notice that $ V $ is a positive and increasing function, therefore, $ f $ is decreasing. Hence $ f(x)-f(0)=-V_{0}^{x}) $.
We then have $ V_{0}^{x}=(V_{0}^{x})^{2} $
It means that there is a point $ a $ in $ [0,1] $ such that $ V $ jumps from $ 0 $ to $ 1 $ right after the point. (It has to occur like that in order to fulfill the identity.)
So $ f(x)= f(0) \ \forall x\in[0,a] $ and $ f(x)= f(0)-1 \ \forall x\in(a,1] $. Thus
$ \frac{1}{3}=\int_0^1f(x)dx=af(0) + (1-a)(f(0)-1) $
